Möbius-invariant curve and surface energies and their applications

Curvature-based surface energies are frequently used in mathematics, physics, thin plate and shell engineering, and membrane chemistry and biology studies. Invariance under rotations and shifts makes curvature-based energies very attractive for modeling various phenomena. In computer-aided geometric design, the Willmore surfaces and the so-called minimum variation surfaces (MVS) are widely used for shape modeling purposes. The Willmore surfaces are invariant w.r.t conformal transformations (Mobius or conformal invariance), and studied thoroughly in differential geometry and related disciplines. In contrast, the minimum variation surfaces are not conformal invariant. In this paper, we suggest a simple modification of the minimum variation energy and demonstrate that the resulting modified MVS enjoy Mobius invariance (so we call them conformal-invariant MVS or, shortly, CI-MVS). We also study connections of CI-MVS with the cyclides of Dupin. In addition, we consider several other conformal-invariant curve and surface energies involving curvatures and curvature derivatives. In particular, we show how filtering with a conformal-invariant curve energy can be used for detecting salient subsets of the principal curvature extremum curves used by Hosaka and co-workers for shape quality inspection purposes.